DOI 10.1393/ncc/i2011-10973-y
Colloquia_{: Channeling 2010}

**Pair photoproduction in a constant and homogeneous**

**electromagnetic ﬁeld**

V. M. Katkov_{(∗)}

*Budker Institute of Nuclear Physics - 630090 Novosibirsk, Russia*

(ricevuto il 22 Dicembre 2010; pubblicato online il 21 Settembre 2011)

**Summary. —** The process of pair photoproduction in an electromagnetic ﬁeld
of arbitrary conﬁguration is investigated. At high energy the correction to the
standard quasiclassical approximation (SQA) of the process probability has been
calculated. In the region of intermediate photon energies where SQA is inapplicable
the new approximation is used. The inﬂuence of weak electric ﬁeld on the process
in a magnetic ﬁeld is considered. In particular, in the presence of this ﬁeld the
root divergence in the probability of pair creation on the Landau energy levels is
vanished. For smaller photon energies the low energy approximations have been
derived. At very low photon energy the found probability describes the absorption
of soft photon by the particles created by the ﬁeld. At low photon energy the electric
ﬁeld action dominates and the inﬂuence of the magnetic ﬁeld on the process occurs
because of its interaction with the magnetic moment of creating particles.

PACS 12.20.-m – Quantum electrodynamics. PACS 13.60.Le – Meson production.

**1. – Introduction**

The pair photoproduction in an electromagnetic ﬁeld is the basic QED reaction which
can play a signiﬁcant role in many processes. This process was considered ﬁrst in a
magnetic ﬁeld. The investigation was started in 1952 independently by Klepikov and
Toll [1,2]. In Klepikov’s paper [3], which was based on the solution of the Dirac equation,
*the probability of photoproduction had been obtained on the mass shell (k*2*= 0, k is the*
4-momentum of photon. We use the system of units with * = c = 1 and the metric*

*ab = aμ _{bμ}*

*0*

_{= a}*0*

_{b}*operator in a magnetic ﬁeld using the proper-time technique developed by Schwinger [5] and Batalin and Shabad [6] had calculated this operator in an electromagnetic ﬁeld using*

_{− ab). In 1971 Adler [4] had calculated the photon polarization}*(∗) E-mail: V.M.Katkov@inp.nsk.su*

c

the Green’s function found by Schwinger [5]. In 1975 the contribution of charged-particles
*loop in an electromagnetic ﬁeld with n external photon lines had been calculated in [7].*
*For n = 2 the explicit expressions for the contribution of scalar and spinor particles*
to the polarization operator of photons were given in this work. Making use of the
imaginary part of this operator for spinor particles the pair photoproduction probability
was analyzed in the pure magnetic [8] and the pure electric [9] ﬁeld.

The probability of pair photoproduction in a constant and homogeneous electric ﬁeld in the quasi-classical approximation had been found by Narozhny [10] using the solution of the Dirac equation in the Sauter potential [11]. Nikishov [12] had obtained the diﬀerential distribution of this process also using the solution of the Dirac equation in the indicated ﬁeld.

In the present paper we consider the integral probability of pair creation by an
un-polarized photon in a constant and homogeneous electromagnetic ﬁeld of an arbitrary
**conﬁguration basing on the polarization operator [7]. In sect. 2 the exact expression**
*for this probability has been received for the general case k*2 *_{= 0. In sect. 3 the }*
stan-dard quasi-classical approximation (SQA) [13, 14] is outlined for the high-energy photon

*ω m (m is the electron mass). The corrections to SQA, determined also the *

applica-bility region of SQA, have been calculated. The found expressions, given in the Lorentz
**invariant form, contain two invariant parameters. In sect. 4 the new approach has been**
developed for the relatively low energies where SQA is not applicable. This approach
is based on the method proposed in [8]. The obtained probability is valid in the wide
**interval of photon energy, which is overlapped with SQA. In sect. 5 the case of the **
*“non-relativistic” photon ω* * m is analyzed. In particular, in the energy region ω eE/m*
where the previous approach is inapplicable, the low energy and the very low energy
approximations have been developed basing on the analysis in [9]. In turn the found
results have an overlapping region of applicability with the previous approach and with
each other. So just as in [9] we have four overlapping approximations which include all
*photon energies. At the photon energy ω eEm/(m*2*+ eE) the probability has been*
*found for arbitrary values of electric E and magnetic B ﬁelds.*

**2. – General expressions for the probability of process**

Our analysis is based on the general expression for the contribution of spinor particles
to the polarization operator obtained in a diagonal form (Baier, Katkov, Strakhovenko,
*1975). The imaginary part of the eigenvalue κiof this operator on the mass shell (k*2= 0)

*determines the probability per unit length Wiof the e−e*+pair creation by the real photon

*with the polarization ei*directed along the corresponding eigenvector. The consideration

**realizes in the frame where the electric E and magnetic B ﬁelds are parallel and directed**
along the axis 3. The probability of pair creation by the unpolarized photon has the
form
*W =* *αm*
2_{r}*2πiω* *μν*
1
*−1* *dv*
_{∞−i0}*−∞−i0* *f (v, x) exp[iψ(v, x)]xdx;*
(1)
*r =* *ω*
2* _{− k}*2
3

*4m*2

*,*

*ν =*

*eE*

*m*2 =

*E*

*E*0

*,*

*μ =*

*eB*

*m*2 =

*B*

*B*0

*,*(2)

*E*0

*= 1.32· 10*16

*V/cm,*

*B*0

*= 4.41· 10*13

*G.*

Here
*f (v, x) =* *cosh(νx)(cos(μx)− cos(μxv))*
*sinh(νx) sin*3*(μx)* +
*cos(μx)(cosh(νx)− cosh(νxv))*
*sin(μx) sinh*3*(νx)* *,*
(3)
*ψ(v, x) = 2r*
*cosh(νx)− cosh(νxv)*
*ν sinh(νx)* +
*cos(μx)− cos(μxv)*
*μ sin(μx)*
*− x.*
(4)

After all calculations have been fulﬁlled we can return to a covariant form of the process description using the following expressions:

(5) *ν*2*− μ*2= 2*F =* **E**
2
*E*2
0
*−***B**2
*B*2
0
*,* *νμ =G =* **EB**
*E*0*B*0
*.*

*The SQA is valid for ultrarelativistic created particles (r* * 1) and can be derived*
*from eqs. (1)-(4) by expanding the functions f (v, x), ψ(v, x) over x powers. Retaining*
*the leading powers of x one obtains*

*W*(SQA)= *αm*
2
3*√3πω*
1
0
9*− v*2
1*− v*2*K2/3(z)dv,* *z =*
8
3(1*− v*2* _{)κ},*
(6)

*κ*2

*= 4r(μ*2

*+ ν*2) =

*−e*2

*m*6

*(F*

*μν*

_{kν}_{)}2

*.*

*To get the correction to the probability in SQA we shall keep the next powers of x. We*
have
(7) *W*(1)=*−* *αm*
2* _{F}*
30

*√3πωκ*1 0

*dv*1

*− v*2

*G(v, z),*where (8)

*G(v, z) = 2(1 + v*2

*− 27z*2

*)K1/3(z) + 3(7− v*2

*)zK2/3(z).*

It is seen that in this order of decomposition the correction does not depend on the
invariant parameter*G, because G is the pseudoscalar. The asimptotics of the integrals*
incoming in the correction terms have been given in Appendix C [8]. The asymptotic at

*κ 1 will become necessary further*

(9) *W*(1)= *6αm*
2_{F}*5ωκ*2
2
3exp
*−* 8
*3κ*
*,* *W*
(1)
*W*(SQA) =
64*F*
*15κ*3*.*
**3. – Region of intermediate photon energies**

*In the ﬁeld, which is weak in comparison with the critical ﬁeld E/E*0 *= ν* * 1,*

*B/B*0*= μ 1 and at the relatively low photon energies r ν−2/3*, the standard
*quasi-classical approximation is non-applicable. At these energies, if the condition r* * ν*2 _{is}
*fulﬁlled, the saddle-point method can be applied to integration over x. In this case the*

*small values of v contribute to the integral over v. So one can expand the phase ψ(v, x)*
*over v and extend the integration limit to the inﬁnity. We get*

(10) *W =* *αm*
2_{r}*2πiω* *μν*
_{∞}*−∞*
*dv*
_{∞}*−∞*
*f (0, x) exp**−i**ϕ(x) + v*2*χ(x)* *xdx,*
where
*ϕ(x) = 2r*
1
*μ*tan
*μx*
2 *−*
1
*ν* tanh
*νx*
2
*+ x,*
(11)
*χ(x) = rx*2
*ν*
*sinh(νx)* *−*
*μ*
*sin(μx)*
*.*
(12)

*From the equation ϕ(x*0*) = 0 we ﬁnd the saddle point x*0

(13) tan2*νs*
2 + tanh
2*μs*
2 =
1
*r,* *x*0=*−is.*
*At r 1 we have*
(14) *W =* *3αm*
2_{κ}*16ω*
3
2exp
*−* 8
*3κ*+
64*F*
*15κ*3
*.*

*This expression is valid at κ* * 1 and coincides with eq. (9) for F κ*3_{.} _{So the}
overlapping region of both approximations exists.

It is interesting to consider the photon energy region*|r − 1| 1 in the presence of a*
*weak electric ﬁeld (ν* * μ) where in the absence of an electric ﬁeld the approach under*
*consideration is valid if the condition r− 1 μ is fulﬁlled. In this case the following*
approximate equations are valid:

*ξ*2* _{y}*2
0
16

*exp[−y*0] + 1

*− r*4

*,*

*y*0

*= μs, ξ =*

*ν*

*μ*; (15)

*y*0

*2 ln*2

*ξ ln*4

*1*

_{ξ}*−*

*r− 1*

*2ξ*2

_{ln}2

*ξ*ln 3 4

*ξ*

*,*

*|r − 1| ξ*2; (16)

*y*0

*ln*4

*r− 1*1

*−*

*ξ*2

*4(r− 1)*ln 4

*r− 1*

*,*

*r− 1 ξ*2; (17)

*ξy*0

*= νs 2*

*√*1

*− r, 1 − r ξ*2

*.*(18)

The applicability of the using saddle-point method is connected with the large value of
*the coeﬃcient to the second power (y− y*0)2 *of the decomposition in the phase ϕ(x). In*
the energy region under consideration we have

(19) *iϕ(x*0*)(x− x*0)2*/2*
*ξ*2
*4μ*
*y*0+
*y*2
0
2 +
*2(r− 1)*
*ξ*2
*(y− y*0)2*.*

*So, in the case ν/μ = ξ* * 1, |r − 1| ξ*2 _{this approximation is valid if the condition}

be available for that. And in the case 1 * 1 − r ξ*2 the condition *√*1*− rξ/μ =*
*(ξ*2_{/μ)(1}− r)/μ 1 is necessary.*At energy r 1 (ν*2* r ν2/3) we have νs π − 2√r,*
(20) *W =* *αm*
2_{μ}*4ω√rcoth(πη) exp*
*−*1
*ν*
*π− 4√r +2r*
*η* tanh
*πη*
2
*.*

*At η 1 the probability W has been increased by the factor ηπ exp[πr/ν] in comparison*
with the case of the absence of magnetic ﬁeld.

**4. – Approximations at low photon energy**

*At r* *∼ ν*2 the above approximation becomes non-applicable and another approach
*has to be applied. We close the integration over x contour in the lower half-plane and*
represent eq. (1) in the following form:

(21) *W =* *αm*
2_{r}*2πiω* *μν*
1
*−1*
*dv*
*∞*
*n=1*
*f (v, x) exp[iψ(v, x)]xdx,*

where the path of integration is any simple closed contour around the point*−iπnν. Let*
*us choose the contour near this point in the following way νx =−iπn+ξn*,*|ξn| ∼√r∼ ν*

*and expand the function entering in over the variables ξn. In the case ν* * 1, because of*

the appearance of the factor exp[*−iπnν], the main contribution to the sum gives the*
*term n = 1. Near the point−iπν the main terms of expansion (ξ ≡ ξ*1) are

(22) *f =* 2i
*ξ*3*coth(πη) cos*
2*πv*
2 *,* *ψ =*
*4r*
*ξν*cos
2*πv*
2 *−*
*ξ*
*ν* +
*iπ*
*ν.*

*Using eq. (7.3.1) and eq. (7.7.1 (11)) in [15] we ﬁnd after integration over ξ and v*

(23) *W =* *αm*
2
*ω* *ηπ coth(πη) exp*
*−π*
*ν*
*I*_{1}2*(z),* *z =* 2
*√*
*r*
*ν* *,*

*where In(z) is the Bessel function of the imaginary argument. The found probability is*

*applicable for r ν.*

*For r ν*2 _{the asymptotic representation I}

*n(z) exp[z]*
*√*

*2πz can be used. As a*
result one obtains eq. (20) if in the exponent of the last one leaves out the term*∝ r/ν.*
*At very low photon energy r* * ν*2, using the expansion of the Bessel functions for the
small value of argument, we have

(24) *W =* *αm*
2_{r}*ων*2 *ηπ coth(πη) exp*
*−π*
*ν*
*.*

The probability under consideration draws the interest of theoreticians for arbitrary
*values μ and ν. For r ν*2*(1 + ν*2*) one can conserve in the phase ψ(v, x) the term−x*

*only. After integrating over v we get the following equation for the probability of photon*
absorption:
*W =* *αm*
2_{r}*iπω*
*∞*
*n=1*
*F (yn*) exp
*−iyn*
*ν*
*dyn,* *yn*=*−inπ + y,*
(25)

*F (y) =* *cosh(y) (ηy cos(ηy)− sin(ηy))*

*sinh y sin*3*ηy* +

*η cos(ηy)(y cosh y− sinh y)*

sinh3*y sin(ηy)* *.*

(26)

*Summing the residues in the points yn*=*−inπ one obtains*

*W =* *αm*
2_{r}*ω*
*∞*
*n=1*
exp
*−πn*
*ν*
*Φ(zn),* *zn= ηπn,*
(27)
*Φ(zn*) =
*zn*
*ν*2*coth zn*+
2
sinh2*zn*
_{ηz}*n*
*ν* +
*1 + η*2*zncoth zn− 1*
*.*
(28)

*At η→ 0, zn→ 0 we have (compare with eq. (28) in [9])*
Φ = 1
*ν*2 +
2
*νπn*+
2
*π*2* _{n}*2 +
2
3

*,*(29)

*W =*

*αm*2

_{r}*ω*1

*ν*2 + 2 3 1

*eπ/ν*

_{− 1}*−*2

*πν*ln 1

*− e−π/ν*+ 2

*π*2Li2

*e−π/ν*

*,*(30)

where Li2*(z) is the Euler dilogarithm. In the opposite case η→ ∞, zn* *→ ∞ one obtains*

(31) Φ = *πηn*
*ν*2 *,* *W =*
*αm*2*r*
*ων*2
*πη*
4 sinh
*−2* *π*
*2ν.*
**5. – Conclusion**

The probability of the process has been calculated using four diﬀerent overlapping approximations. In the region of SQA applicability the particles created by a photon have ultrarelativistic energies. The role of ﬁelds in this case is to transfer the required transverse momentum and the electric and magnetic ﬁeld actions are equivalent. But even in this case it is necessary to note the special signiﬁcance of the weak electric ﬁeld

*E = ξB (ξ* * 1) in the removal of the root divergence of the probability when the*

particles of the pair are created on the Landau levels with the electron and positron
*momentum p*3*= 0. In the frame used, k*3= 0.

*Generally speaking, at ξ 1 the formation time tc*of the process under consideration
*is 1/μ. Here we use units = c = m = 1. At this time the particle of the pair gets the*
*momentum δp*3*∼ ξ because of the electric ﬁeld. If the value ξ*2becomes more larger than
*the distance apart Landau levels 2μ (ν*2 * _{ μ}*3

_{) all levels have been overlapped. Under}this condition the divergence of the probability is vanished and the new quasi-classical

*approach is valid even in the energy region r−1 μ where it has been inapplicable in the*

*absence of the electric ﬁeld. In the opposite case ν*2

*μ*3

*for the small value of p*3

*√μ*in the region where the inﬂuence of the electric ﬁeld can be neglected, the formation time

*of the process tf*

*is 1/p*23

*and δp*3

*∼ ν/p*23

*p*3. It follows from the above that in this

*case the condition ν1/3*

*p*3

*√μ has to be satisﬁed. At this condition the value of*

discontinuity is *tf/tc* *∼ √μ/p*3*. For ν1/3 p*3 *the time tf* is determined by the

*self-consistent equation δε*2*∼ 1/tf* *∼ ν*2*t*2_{f}, tf*∼ ν−2/3*and the value of discontinuity becomes
*√*

*μtf* *∼ (μ*3*/ν*2)*1/6* instead of*√μ/p*3*. In the region ω 2m (r 1) the energy transfer*
*from the electric ﬁeld to the created particles becomes appreciable and for ω* * m it*
*determines mainly the probability of the process. At ω eE/m the photon assistance*
in the pair creation comes to an end and the probability under consideration deﬁnes
the probability of photon absorption by the particles created by electromagnetic ﬁelds.
The inﬂuence of the magnetic ﬁeld on the process is connected with the interaction of
the magnetic moment of the created particles and magnetic ﬁeld. This interaction, in
particular, has appeared in the distinction of the pair creation probability by ﬁeld for
scalar and spinor particles [5].

*∗ ∗ ∗*

This work was partly supported by Grant 14.740.11.0082 of Federal Program “Per-sonnel of Innovational Russia”.

REFERENCES

[1] Klepikov N. P., Ph.D. dissertation, unpublished (Moscow State University) 1952. [2] Toll J. S., Ph.D. dissertation, unpublished (Prinston University) 1952.

**[3] Klepikov N. P., Zh. Eksp. Teor. Fiz., 26 (1954) 19.****[4] Adler S. L., Ann. Phys. (N.Y.), 67 (1971) 599.****[5] Schwinger J., Phys. Rev., 82 (1951) 664.**

**[6] Batalin I. A. and Shabad A. E., Sov. Phys. JETP, 33 (1971) 483.**

* [7] Baier V. N., Katkov V. M. and Strakhovenko V. M., Sov. Phys. JETP, 41 (1975)*
198.

**[8] Baier V. N. and Katkov V. M., Phys. Rev. D, 75 (2007) 073009.****[9] Baier V. N. and Katkov V. M., Phys. Lett. A, 374 (2010) 2201.****[10] Narozhny N. B., Zh. Eksp. Teor. Fiz., 54 (1968) 676.**

**[11] Sauter F., Z. Phys., 69 (1931) 742.**

**[12] Nikishov A. I., Zh. Eksp. Teor. Fiz., 59 (1970) 1262.**

**[13] Baier V. N. and Katkov V. M., Sov. Phys. JETP, 26 (1968) 854.****[14] Tsai W. and Erber T., Phys. Rev. D, 10 (1974) 492.**

[15] Bateman H. and Erd´elyi A.*_{, Higher Transcendental Functions, Vol. II (McGraw-Hill}*
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